Stable radiation-controlling boundary conditions for the generalized harmonic Einstein equations

Abstract

This paper is concerned with the initial-boundary value problem for the Einstein equations in a first-order generalized harmonic formulation. We impose boundary conditions that preserve the constraints and control the incoming gravitational radiation by prescribing data for the incoming fields of the Weyl tensor. High-frequency perturbations about any given spacetime (including a shift vector with a subluminal normal component) are analysed using the Fourier–Laplace technique. We show that the system is boundary stable. In addition, we develop a criterion that can be used to detect weak instabilities with polynomial time dependence, and we show that our system does not suffer from such instabilities. A numerical robust stability test supports our claim that the initial-boundary value problem is most likely to be well posed even if non-zero initial and source data are included.